York E. Young, Department of Physics and Astronomy
Aware of the large number of insights which may result from a better understanding of non-neutral plasmas, the Brigham Young University plasma group currently explores the electrostatic normal modes of oscillation in confined plasmas. Researchers at BYU currently perform computational studies of waves In non-neutral plasmas confined in real geometries in support of experiments both here and elsewhere.
Most numerical studies utilize particle simulation codes. However, several features of a plasma’s dynamics are difficult to observe with such codes.
Another approach Is to treat the plasmas as fluids, applying fluid equations and theory to the description of their dynamics. One drawback with this method is that numerical methods for solving fluid equations often exhibit what is known as “ringing.” As a plasma moves in a trap, regions of high density meet regions of low density. Most fluid equation algorithms respond to such steep discontinuities by introducing small wavelength oscillations into the solution. This results in the prediction of negative densities, which Is known as ringing and is of course aphysical.
To overcome this problem Rachel Berg, a former BYU student, has made use of Colella and Woodward’s piecewise parabolic method to numerically solve the convection problem’. This algorithm does not allow ringing In the presence of steep discontinuities and is second order accurate in such regions. Additionally it has the advantages of being stable, non-diffusive, and fourth order accurate in regions where the solution is smooth.
A particle simulation of non-neutral plasmas produces damping In the modes of oscillation, but the noisiness of the results led researchers to question the physicality of this damping. The aim of this research was to create a simulation program which models a plasma by convecting Its particle density In phase space rather than moving Its particles Individually, with the Intent that this would yield much smoother results and lend some insight Into the physicality and cause of the somewhat questioned damping.
First the Piecewise Parabolic Method’ (hereafter referred to as PPM) Is used to obtain a solution, a (~,t), to the constant velocity homogeneous convection equation In one spatial dimension (~)
The method involves a careful choice of an Interpolating polynomial which eliminates the Introduction of spurious oscillations Into the solution, avoiding a problem arising frequently In other methods.
Secondly, a numerical method capable of solving the Vlasov equation
(a two-dimensional convection equation), was accomplished by means of operator splitting’, enabling the one-dimensional PPM to solve the two-dimensional Vlasov equation.
And finally, this algorithm was placed Into a program called “Snake,” which simulates the behavior of a trapped plasma by convecting its particle density In phase space. Snake’s utility Is made manifest by comparing its results with those of a program called “Rattle,” which models a plasma’s behavior in the quite different manner of moving its particles according to Newton’s second law. Snake’s results are much smoother and provide greater qualitative insight Into what causes the damping. Finally, the fact that both very Independent programs produce the same damping suggests that It Is a truly physical phenomenon.
For detailed information regarding this research see reference 4.
References
- R.K. Berg, Solution of the Convection Equation by the Piecewise Parabolic Method, Honors Thesis, Brigham Young University. 1994.
- P. Colella and P.R. Woodward, Journal of Camp. Phys. 54, pp. 174-181. (1984).
- W.H. Press, B.P. Flannery, S.A. Teulosky, and W.T. Vetterling, Numerical Recipes (Cambridge U.P., Cambridge) pp. 660-667 (1986).
- Y.E. Young, The Piecewise Parabolic Method in Plasma Simulation, Honors Thesis, Brigham Young University. 1995.